Second-order vibronic reduction factors for orbital triplet Jahn–Teller systems in cubic and icosahedral symmetry

Abstract

It is well-known that vibronic interactions can be modelled in terms of an effective Hamiltonian incorporating first and second-order reduction factors (RFs), particularly when analysing the spectroscopic properties. Measurements and calculations of the RFs as a function of the strength of vibronic coupling are therefore of much interest. In this paper, we develop a new general method for determining second-order RFs (soRFs) from the strength of the Jahn–Teller (JT) coupling for systems in which electron orbital degeneracy or pseudo-degeneracy exists. These include in particular the fullerene molecule C60, pseudo-Jahn–Teller molecules and impurity centres in crystals. In order to calculate the important soRFs for intermediate to strong coupling, it is necessary to determine non-Condon corrections to the strong coupling values obtained using the Franck–Condon (FC) approximation. This gives an additional contribution to the nuclear polarizability of the system, thus enabling the electrons to follow the nuclear vibrations. These non-Condon corrections are derived using perturbation theory and are found to be inversely proportional to the square of the JT energy. The validity of the approximation is first tested in the cubic JT system due to its relative simplicity. It is found that the results are closer to those obtained earlier by numerical methods than the analytical FC values alone. Results are then presented that are applicable to C60− anions.

Accurate calculations of second-order vibronic reduction factors for C60ions

“…in the present study involving the non-Condon modification of FC calculations. A comparison between the two approaches has already been made for the T ⊗ h JT system which has a direct relevance to the ground state of the C − 60 molecular anion [6]. Nevertheless, it is difficult to compare directly the new sets of results obtained here with our previous results as the FC approximation is based on an infinite strength coupling model.…”

“…Electronic perturbations generate first-and second-order Hamiltonians describing the coupling within the orbital state of symmetry , as discussed in detail in [6] and [9], for example. The definition of the soRFs K (2) M ( l ⊗ m ) involving two perturbations, H (1) ( l ) of symmetry l and H (1) ( m ) of symmetry m , can be found in the general theory summarized in [10].…”

“…As discussed in [6], the effect of the non-Condon (NC) terms is to give a correction | ψ 1 p to the ground state in well p which is proportional to the well displacement coordinate q p = Q j − Q 0 p . (This correction arises from the term U 1 in the corrected potential energy given in equation ( 13) of [6].) Then the coordinate Q j replaces Q γ in equation (1).…”

“…The major problem is that soRFs are generally difficult to calculate with high accuracy, except in the limit of infinitely strong coupling when the Franck-Condon (FC) approximation [5] is valid. However, in order to overcome many of these difficulties, a new procedure has been devised involving non-Condon corrections to the standard FC method [6]. This extends the range of applicability of the results to much lower coupling strengths.…”

Anisotropic effect on the vibronic reduction factors

“…These factors can describe the energy splitting caused by magnetic field, strain and hyperfine effects because the reduction factor can not only reflect the influence of JT effect of the crystal on the spectrum, but also sufficiently describe the JT effect on some weak-perturbed systems. Therefore, many authors carried on the research of the reduction factors [2][3][4][5] of JT systems, including the icosahedral systems related to C 60 molecules. The experiments proved that the vibronic coupling have a quenching effect on the spin-orbit couplings, random strains and internal stresses, etc., and these effects can be explained by reduction factors.…”